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Mirrors > Home > MPE Home > Th. List > hlcomd | Structured version Visualization version GIF version |
Description: The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
ishlg.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
hlcomd.1 | ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) |
Ref | Expression |
---|---|
hlcomd | ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcomd.1 | . 2 ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) | |
2 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
5 | ishlg.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | ishlg.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | ishlg.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | ishlg.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
9 | 2, 3, 4, 5, 6, 7, 8 | hlcomb 26391 | . 2 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐵(𝐾‘𝐶)𝐴)) |
10 | 1, 9 | mpbid 234 | 1 ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 Itvcitv 26224 hlGchlg 26388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-hlg 26389 |
This theorem is referenced by: hlcgreulem 26405 opphllem4 26538 opphllem5 26539 opphl 26542 hlpasch 26544 lnopp2hpgb 26551 colhp 26558 cgrahl1 26604 cgrahl2 26605 cgrahl 26615 cgracol 26616 dfcgra2 26618 sacgr 26619 acopy 26621 acopyeu 26622 inaghl 26633 tgasa1 26646 |
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